[unreadable] [unreadable] Cryogenic Electron Microscopy (CEM) is a tool for obtaining three dimensional structure of proteins and [unreadable] other biological macromolecules. CEM produces noisy two dimensional images of the structure at random orientations, and the image processing challenge is to produce estimates of projections in known directions by aligning and averaging the images. These estimates (or averages) are constrained because they have to satisfy the so-called common line constraints. [unreadable] [unreadable] This application proposes a constrained maximum-likelihood algorithm that is guaranteed to produce [unreadable] aligned averages that satisfy the common line constraints. The algorithm uses a modified Expectation- [unreadable] Maximization (EM) technique to maximize the likelihood. The modification consists of ensuring that the [unreadable] common lines constraints are satisfied within each EM iteration. The constraints are shown to be linear and are enforced using the von Neuman alternating projections theorem. The algorithm is provably convergent, monotonically improves the likelihood, and is optimal. The mathematical theory and preliminary simulations are presented in the application. [unreadable] [unreadable] The goal of the application is to conduct a through pilot study of the efficacy of the algorithm. The accuracy and resolution of reconstruction will be studied by Monte Carlo simulations with synthetic and real 3-D protein and other macromolecular structures. The accuracy will be measured by the average mean squared error in reconstruction, and resolution will be measured by Fourier shell correlation. Further, the performance of the algorithm will be measured by reconstructing the IP3 Ca ion channel protein. The structure of this protein has been analyzed previously, so a comparison with published structure should provide a good indication of relative accuracy. [unreadable] [unreadable] The ultimate relevance of this research is that it would provide a principled algorithm with guaranteed [unreadable] properties for reconstructing the 3-D structure of proteins. [unreadable] [unreadable] [unreadable]